Topological K-(co-)homology of classifying spaces of discrete groups

TL;DR

This paper develops methods to compute topological K-(co-)homology of classifying spaces of discrete groups, providing formulas applicable to various groups and introducing new tools like a Cocompletion Theorem and Equivariant Universal Coefficient Theorems.

Contribution

It introduces novel computational formulas and tools for topological K-(co-)homology of classifying spaces of discrete groups, extending known results to broader classes of groups.

Findings

Formulas for K-(co-)homology of BG up to finite abelian torsion.

Application to arithmetic, hyperbolic, and mapping class groups.

Introduction of a Cocompletion Theorem and Equivariant Universal Coefficient Theorems.

Abstract

Let G be a discrete group. We give methods to compute for a generalized (co-)homology theory its values on the Borel construction (EG x X)/G of a proper G-CW-complex X satisfying certain finiteness conditions. In particular we give formulas computing the topological K-(co)homology of the classifying space BG up to finite abelian torsion groups. They apply for instance to arithmetic groups, word hyperbolic groups, mapping class groups and discrete cocompact subgroups of almost connected Lie groups. For finite groups G these formulas are sharp. The main new tools we use for the K-theory calculation are a Cocompletion Theorem and Equivariant Universal Coefficient Theorems which are of independent interest. In the case where G is a finite group these theorems reduce to well-known results of Greenlees and Boekstedt.